CN105912793A - Finite element method obtaining hypoid gear teeth bending deformation - Google Patents

Abstract

The invention relates to a finite element method obtaining hypoid gear teeth bending deformation; the method comprises the following steps: 1, restraining wheel blank inner ring node all degrees of freedom on a pinion finite element model, thus obtaining a pinion model I; 2, further restraining pinion tooth thickness central section intermediate node circumferential freedom degrees, thus obtaining a pinion model II; 3, calculating tooth flank coacervation rigidity matrixes KP1 and KP2 of the pinion models I and II; 4, restraining wheel blank inner ring node all degrees of freedom on a bull gear finite element model, thus obtaining a bull gear model I; 5, further restraining bull gear tooth thickness central section intermediate node circumferential freedom degrees, thus obtaining a bull gear model II; 6, calculating tooth flank coacervation rigidity matrixes KG1 and KG2 of the bull gear models I and II; 7, applying unit normal load in sequence on all points of a bull gear-pinion contact line, and calculating pinion and bull gear tooth flank normal flexibility matrixes RP and RG of each contact line; 8, using RP and RG to calculate bending deflection corresponding to each point according to real load F on the contact points.

Description

A kind of acquisition diastrophic Finite Element Method of the hypoid gear gear teeth

Technical field

The present invention relates to a kind of method obtaining tooth bending deformation, obtain hypoid gear wheel especially with regard to one The diastrophic Finite Element Method of tooth.

Background technology

Hypoid gear loaded tooth contact analysis (LTCA) method is a key in hypoid gear design process Technology.Its Main Function is, before carrying out cutting with actual machined parameters, first calculates hypoid gear and carries in real work Each performance indications under lotus, load contact trace including the flank of tooth and load driving error etc..Utilize LTCA technology can shorten standard The R&D cycle of hypoid gear and R&D costs.

In current LTCA Research on Calculation, the deformation of the gear teeth is broadly divided into three parts, respectively flexural deformation δ_{Flexural deformation}、 Juxtaposition metamorphose δ_{Juxtaposition metamorphose}With detrusion δ_Detrusion(as shown in Figure 1).Krenzer, Zheng Changqi etc. (list of references: Krenzer, T.J.Tooth contact analysis of spiral and hypoid gears under load.The Gleason Works, SAE810688,1981 and Zheng Changqi, spiral bevel gear loaded Tooth Contact Analysis principle. mechanical engineering journal, 1993.29 (4)) using Hertz contact formula to calculate the juxtaposition metamorphose of the flank of tooth, flexural deformation and detrusion have employed consideration and cuts Cut the Flexural cantilever model of impact.

Owing to Flexural cantilever model more simplifies, overall precision is the highest, and Wilcox, Fan Qi etc. proposes to use Weber experience Formula calculates juxtaposition metamorphose and detrusion；Use gear teeth solid finite meta-model calculating flexural deformation (list of references: Wilcox,L.E.Improved Finite Element Model for Calculating Stresses in Bevel And Hypoid Gear Teeth.AGMA, 97FTM5,1997 and Q.Fan, L.Wilcox, New Developments in Tooth Contact Analysis(TCA)and Loaded TCA for Spiral Bevel and Hypoid Gear Drives.AGMA,05FTM08,2005.)。

In above-mentioned use gear teeth solid finite meta-model calculates bending deformation process, constrain wheel blank inner ring node whole Degree of freedom.When calculating flexural deformation, it is believed that the normal deformation of the contact point of gear surface obtained is exactly the flexural deformation in the middle part of the gear teeth. This situation is for being to set up using whole tooth and tooth base as rigid body, but actually has due to gear teeth solid finite meta-model Flexibility, further comprises the deformation caused due to flexibility in the flexural deformation therefore obtained；Also contains at load(ing) point and wheel simultaneously Detrusion between base, this part deformation is calculated in weber formula.

Summary of the invention

For the problems referred to above, it is an object of the invention to provide one and can eliminate in gear teeth flexural deformation during calculating The acquisition diastrophic Finite Element Method of the hypoid gear gear teeth of other deformation comprised.

For achieving the above object, the present invention takes techniques below scheme: a kind of hypoid gear gear teeth that obtain bend change The Finite Element Method of shape, comprises the following steps: the 1) FEM (finite element) model to little gear, and constraint wheel blank inner ring node is all freely Degree, the model that this step is set up is referred to as the little model of gear I；2) on the basis of the little model of gear I, in constraint pinion gear teeth thickness The circumferential degree of freedom of portion's intermediate node, is compressed in order to the gear teeth after calculating the applying power caused owing to FEM (finite element) model is non-rigid Detrusion at the compression produced and load(ing) point and between wheel blank, the model that this step is set up is referred to as little gear die Type II；3) utilize the little model of gear I, calculate the flank of tooth cohesion stiffness matrix K under its corresponding restraint condition_P1；Utilize little gear die Type II, calculates the flank of tooth cohesion stiffness matrix K under its corresponding constraint_P2；4) FEM (finite element) model to gear wheel, retrains wheel blank inner ring The whole degree of freedom of node, the model that this step is set up is referred to as gear wheel Model I；5) on the basis of gear wheel Model I, constraint The circumferential degree of freedom of intermediate node in the middle part of gear wheel transverse tooth thickness, after calculating the applying power caused owing to FEM (finite element) model is non-rigid The gear teeth are by compressing the detrusion at the compression and load(ing) point produced and between wheel blank, the model that this step is set up It is referred to as gear wheel model-；6) utilize gear wheel Model I, calculate the flank of tooth cohesion stiffness matrix K under its corresponding restraint condition_G1； Utilize gear wheel model-, calculate the flank of tooth cohesion stiffness matrix K under its corresponding constraint_G2；7) respectively to size Gear Contact line On institute the most successively apply unit normal load, calculate every contact line pinion gear teeth face normal direction flexibility matrix R_PAnd gear wheel Flank of tooth normal direction flexibility matrix R_G；8) little gear and gear wheel flank of tooth normal direction flexibility matrix R are utilized_PAnd R_G, according to actual on contact point Load F, calculates the bending deformation quantity that each point is corresponding: δ according to formula below_{Flexural deformation}=(R_P+R_G)F。

Described step 7) in:

Pinion gear teeth face normal direction flexibility matrix R on any bar contact line_PCalculating process as follows:

Line is contacted for any bar, utilizes the flank of tooth cohesion stiffness matrix K of the little model of gear I_P1, each on contact line successively Point applies unit normal load F_unitAfter, formula (1) obtain each contact point displacement d_P1:

$d_{P1}=\[N_s\]\[K_{P1}^{-1}\]\left\{F_{unit}\right\}---\left(1\right)$

In formula, N_sFor unit shape function, it is used for displacement of joint is transformed into contact point；

Utilize the flank of tooth cohesion stiffness matrix K of the little model of gear II_P2, on contact line, each point applies unit normal direction load successively Lotus F_unitAfter, formula (2) obtain each contact point displacement d_P2:

$d_{P2}=\[N_s\]\[K_{P2}^{-1}\]\left\{F_{unit}\right\}---\left(2\right)$

Utilize the displacement of each contact point that the displacement of each contact point that the little model of gear I obtains obtains with the little model of gear II Subtract each other, i.e. formula (3), obtain this contact line pinion gear teeth face normal direction flexibility matrix R_P:

R_P[i, j]=d_P1-d_P2, (i, j=1 ..., m) (3)

In formula, m is the number of contact point on this contact line；

Gear wheel flank of tooth normal direction flexibility matrix R on any bar contact line_GCalculating process as follows:

Line is contacted for any bar, utilizes the flank of tooth cohesion stiffness matrix K of gear wheel Model I_G1, each on contact line successively Point applies unit normal load F_unitAfter, formula (4) obtain each contact point displacement d_G1:

$d_{G1}=\[N_s\]\[K_{G1}^{-1}\]\left\{F_{unit}\right\}---\left(4\right)$

Utilize the flank of tooth cohesion stiffness matrix K of gear wheel model-_G2, on contact line, each point applies unit normal direction load successively Lotus F_unitAfter, formula (5) obtain each contact point displacement d_G2:

$d_{G2}=\[N_s\]\[K_{G2}^{-1}\]\left\{F_{unit}\right\}---\left(5\right)$

Utilize the displacement of each contact point that the displacement of each contact point that gear wheel Model I obtains obtains with gear wheel model- Subtract each other, i.e. formula (6), obtain this contact line gear wheel flank of tooth normal direction flexibility matrix R_G:

R_G[i, j]=d_G1-d_G2, (i, j=1 ..., m). (6)

Due to the fact that and take above technical scheme, it has the advantage that 1, utilize that the inventive method obtains accurate double Curved surface gear tooth flexural deformation, its result is not mixed has other to deform.2, the inventive method obtain loading contact trace with The degree of agreement of actual loaded experimental result, the identical journey loading contact trace and loading experiment result obtained than traditional method Spend.

Accompanying drawing explanation

Fig. 1 is three kinds of deformation schematic diagrams in hypoid gear loaded tooth contact analysis method；

Fig. 2 is the way of restraint two-dimensional representation that in the inventive method, Model I is corresponding；

Fig. 3 is the way of restraint two-dimensional representation that in the inventive method, model- is corresponding；

Fig. 4 is that in the inventive method, every contact collimation method calculates process schematic to flexibility matrix；

Fig. 5 is the loading contact trace figure that in specific embodiment, hypoid gear pair uses the inventive method to obtain；

Fig. 6 is the loading contact trace figure that in specific embodiment, hypoid gear pair uses traditional method to obtain；

Fig. 7 is the loading contact trace figure that in specific embodiment, hypoid gear pair carries out that loading experiment obtains.

Detailed description of the invention

With embodiment, the present invention is described in detail below in conjunction with the accompanying drawings.

The present invention proposes a kind of acquisition diastrophic Finite Element Method of the hypoid gear gear teeth, including following step Rapid:

1) FEM (finite element) model to little gear, the constraint whole degree of freedom of wheel blank inner ring node (as shown in Figure 2), this step institute The model set up is referred to as the little model of gear I.

2) on the basis of the little model of gear I, in the middle part of constraint pinion gear teeth thickness, the circumferential degree of freedom of intermediate node is (such as Fig. 3 Shown in), in order to the gear teeth after calculating the applying power caused owing to FEM (finite element) model is non-rigid by the compression that produces of compression with And the detrusion at load(ing) point and between wheel blank, the model that this step is set up is referred to as the little model of gear II.

3) utilize the little model of gear I, calculate the flank of tooth cohesion stiffness matrix K under its corresponding restraint condition_P1；Utilize little gear Model-, calculates the flank of tooth cohesion stiffness matrix K under its corresponding constraint_P2。

4) FEM (finite element) model to gear wheel, retrains the wheel blank whole degree of freedom of inner ring node, the model that this step is set up It is referred to as gear wheel Model I.

5) on the basis of gear wheel Model I, in the middle part of constraint gear wheel transverse tooth thickness, the circumferential degree of freedom of intermediate node is (such as Fig. 3 Shown in), in order to the gear teeth after calculating the applying power caused owing to FEM (finite element) model is non-rigid by the compression that produces of compression with And the detrusion at load(ing) point and between wheel blank, the model that this step is set up is referred to as gear wheel model-.

6) utilize gear wheel Model I, calculate the flank of tooth cohesion stiffness matrix K under its corresponding restraint condition_G1；Utilize gear wheel Model-, calculates the flank of tooth cohesion stiffness matrix K under its corresponding constraint_G2。

7) respectively the institute on size Gear Contact line is applied unit normal load the most successively, calculate every contact line little Gear teeth face normal direction flexibility matrix R_PWith gear wheel flank of tooth normal direction flexibility matrix R_G, its detailed process is as follows:

After gear teeth stand under load, on the flank of tooth, each moment has and contacts line accordingly, the flank of tooth normal direction flexibility square of a certain bar contact line Battle array R [i, j] is defined as being applied specific loading, the deformation of the j caused point by i point on this contact line.Therefore, in order to calculate little gear With the flank of tooth normal direction flexibility matrix of gear wheel, need the point on every contact line is applied corresponding normal direction specific loading successively F_unit, and calculate the deformation of other point being generated by.Here take a contact line therein to illustrate, as shown in Figure 4, On contact line, a contact point 1 applies unit normal force, is leaned on by the unit that this load decomposes this place according to FEM (finite element) model On the node of flank of tooth side, and condense stiffness matrix according to the flank of tooth, be calculated other flank of tooth modal displacements, it is thus achieved that modal displacement After obtain contacting the Normal Displacement of the point 1～4 on line by unit shape function again, i.e. R [1,1]～R [Isosorbide-5-Nitrae], it represents respectively Implication be 1～the deformation of 4 caused by the load of 1.In flank of tooth flexibility matrix R [i, j], other also uses phase Tongfang Method calculates.

The process that utilize Model I and model- seek flank of tooth normal direction flexibility matrix be specifically described below:

1. pinion gear teeth face normal direction flexibility matrix R_P:

Line is contacted for any bar, utilizes the flank of tooth cohesion stiffness matrix K of the little model of gear I_P1, each on contact line successively Point applies unit normal load F_unitAfter, formula (1) obtain each contact point displacement d_P1:

$d_{P1}=\[N_s\]\[K_{P1}^{-1}\]\left\{F_{unit}\right\}---\left(1\right)$

In formula, N_sFor unit shape function, it is used for displacement of joint is transformed into contact point；

Utilize the flank of tooth cohesion stiffness matrix K of the little model of gear II_P2, on contact line, each point applies unit normal direction load successively Lotus F_unitAfter, formula (2) obtain each contact point displacement d_P2:

$d_{P2}=\[N_s\]\[K_{P2}^{-1}\]\left\{F_{unit}\right\}---\left(2\right)$

Utilize the displacement of each contact point that the displacement of each contact point that the little model of gear I obtains obtains with the little model of gear II Subtract each other, i.e. formula (3), obtain this contact line pinion gear teeth face normal direction flexibility matrix R_P:

R_P[i, j]=d_P1-d_P2, (i, j=1 ..., m) (3)

In formula, m is the number of contact point on this contact line.

2. gear wheel flank of tooth normal direction flexibility matrix R_G:

Line is contacted for any bar, utilizes the flank of tooth cohesion stiffness matrix K of gear wheel Model I_G1, each on contact line successively Point applies unit normal load F_unitAfter, formula (4) obtain each contact point displacement d_G1:

$d_{G1}=\[N_s\]\[K_{G1}^{-1}\]\left\{F_{unit}\right\}---\left(4\right)$

Utilize the flank of tooth cohesion stiffness matrix K of gear wheel model-_G2, on contact line, each point applies unit normal direction load successively Lotus F_unitAfter, formula (5) obtain each contact point displacement d_G2:

$d_{G2}=\[N_s\]\[K_{G2}^{-1}\]\left\{F_{unit}\right\}---\left(5\right)$

Utilize the displacement of each contact point that the displacement of each contact point that gear wheel Model I obtains obtains with gear wheel model- Subtract each other, i.e. formula (6), obtain this contact line gear wheel flank of tooth normal direction flexibility matrix R_G:

R_G[i, j]=d_G1-d_G2, (i, j=1 ..., m) (6)

8) little gear and gear wheel flank of tooth normal direction flexibility matrix R are utilized_PAnd R_G, according to real load F on contact point, according to The bending deformation quantity that formula below calculating each point is corresponding:

δ_{Flexural deformation}=(R_P+R_G)F。

Utilize this bending deformation quantity can carry out LTCA analysis, finally obtain hypoid gear loaded tooth contact analysis knot Really.

Below by a specific embodiment, in order to the effect of the present invention to be described.

Taking a pair hypoid gear pair, its fine-processing technique is roll flute, and wherein, little gear concave surface adds at roll flute and adds man-hour The circular arc having entered cutter repaiies type, and arc radius of practicing Buddhism or Taoism is 1500mm.This gear pair design parameter is shown in Table 1:

Table 1 hypoid gear basic parameter table

The flank of tooth information of this gear is determined by machined parameters, and the machined parameters of the gear wheel that this gear mesh is answered and little gear is shown in Table 2 and table 3:

Table 2 bull wheel machined parameters

Table 3 steamboat machined parameters

First, sequentially according to above-mentioned steps 1)～7), try to achieve the little gear every contact little gear of line and gear wheel flank of tooth method To flexibility matrix R_PAnd R_G, given actual operating mode is that positive car drives input torque 1615.4Nm, carries out LTCA iterative analysis, Solve hypoid gear and load contact trace (as shown in Figure 5).Being contact trace cloud atlas on the gear wheel flank of tooth in figure, color is Deep region is not in contact with region, and remainder is contact area.

Then, utilize traditional Finite Element Method to be also carried out LTCA to analyze, it is thus achieved that contact trace accordingly (such as Fig. 6 institute Show).Utilize the hypoid gear pair identical with above-mentioned analysis process to carry out actual loaded experiment, two tooth after being tested simultaneously Contact trace (as shown in Figure 7) during wheel actual loaded engagement, scribbles red lead powder region for not in contact with region, metal zone in figure Territory is actual contact area.The inventive method and traditional method are carried out LTCA analyze the result that obtains and experimental result carry out right Ratio.

By Comparative result it can be seen that the method that the present invention uses is better than traditional method with experimental result degree of agreement. The contact trace expansion rate of traditional method is faster than practical situation, and LTCA analysis result middle gear small end has been deviate from, and big end is opened Begin there is a small amount of abjection；And the inventive method contacts trace at this moment and also has certain distance away from small end, big end the most just reaches abjection Border, coincide more preferably with actual experiment result.

The various embodiments described above are only used for having carried out the purpose of the present invention, technical scheme and beneficial effect the most specifically Bright, be not limited to the present invention, all within the spirit and principles in the present invention, any modification, equivalent substitution and improvement done Deng, should be included within the scope of the present invention.

Claims (2)

1. obtain the diastrophic Finite Element Method of the hypoid gear gear teeth, comprise the following steps:

1) FEM (finite element) model to little gear, retrains the wheel blank whole degree of freedom of inner ring node, and the model that this step is set up is referred to as The little model of gear I；

2) on the basis of the little model of gear I, the circumferential degree of freedom of intermediate node in the middle part of constraint pinion gear teeth thickness, in order to calculate by After the non-rigid applying power caused of FEM (finite element) model, the gear teeth are by compressing at the compression and load(ing) point produced and wheel blank Between detrusion, the model that this step is set up is referred to as the little model of gear II；

3) utilize the little model of gear I, calculate the flank of tooth cohesion stiffness matrix K under its corresponding restraint condition_P1；Utilize the little model of gear II, calculate the flank of tooth cohesion stiffness matrix K under its corresponding constraint_P2；

4) FEM (finite element) model to gear wheel, retrains the wheel blank whole degree of freedom of inner ring node, and the model that this step is set up is referred to as Gear wheel Model I；

5) on the basis of gear wheel Model I, the circumferential degree of freedom of intermediate node in the middle part of constraint gear wheel transverse tooth thickness, in order to calculate by After the non-rigid applying power caused of FEM (finite element) model, the gear teeth are by compressing at the compression and load(ing) point produced and wheel blank Between detrusion, the model that this step is set up is referred to as gear wheel model-；

6) utilize gear wheel Model I, calculate the flank of tooth cohesion stiffness matrix K under its corresponding restraint condition_G1；Utilize gear wheel model II, calculate the flank of tooth cohesion stiffness matrix K under its corresponding constraint_G2；

7) respectively the institute on size Gear Contact line is applied unit normal load the most successively, calculate the every contact little gear of line Flank of tooth normal direction flexibility matrix R_PWith gear wheel flank of tooth normal direction flexibility matrix R_G；

8) little gear and gear wheel flank of tooth normal direction flexibility matrix R are utilized_PAnd R_G, according to real load F on contact point, according to as follows Formula calculate bending deformation quantity corresponding to each point:

δ_{Flexural deformation}=(R_P+R_G)F。

2. a kind of acquisition diastrophic Finite Element Method of the hypoid gear gear teeth as claimed in claim 1, its feature exists In, described step 7) in:

Pinion gear teeth face normal direction flexibility matrix R on any bar contact line_PCalculating process as follows:

Line is contacted for any bar, utilizes the flank of tooth cohesion stiffness matrix K of the little model of gear I_P1, on contact line, each point is executed successively Add unit normal load F_unitAfter, formula (1) obtain each contact point displacement d_P1:

$d_{P1}=\[N_s\]\[K_{P1}^{-1}\]\left\{F_{unit}\right\}---\left(1\right)$

In formula, N_sFor unit shape function, it is used for displacement of joint is transformed into contact point；

Utilize the flank of tooth cohesion stiffness matrix K of the little model of gear II_P2, on contact line, each point applies unit normal load successively F_unitAfter, formula (2) obtain each contact point displacement d_P2:

$d_{P2}=\[N_s\]\[K_{P2}^{-1}\]\left\{F_{unit}\right\}---\left(2\right)$

Utilize the displacement phase of each contact point that the displacement of each contact point that the little model of gear I obtains obtains with the little model of gear II Subtract, i.e. formula (3), obtain this contact line pinion gear teeth face normal direction flexibility matrix R_P:

R_P[i, j]=d_P1-d_P2, (i, j=1 ..., m) (3)

In formula, m is the number of contact point on this contact line；

Gear wheel flank of tooth normal direction flexibility matrix R on any bar contact line_GCalculating process as follows:

Line is contacted for any bar, utilizes the flank of tooth cohesion stiffness matrix K of gear wheel Model I_G1, on contact line, each point is executed successively Add unit normal load F_unitAfter, formula (4) obtain each contact point displacement d_G1:

$d_{G1}=\[N_s\]\[K_{G1}^{-1}\]\left\{F_{unit}\right\}---\left(4\right)$

Utilize the flank of tooth cohesion stiffness matrix K of gear wheel model-_G2, on contact line, each point applies unit normal load successively F_unitAfter, formula (5) obtain each contact point displacement d_G2:

$d_{G2}=\[N_s\]\[K_{G2}^{-1}\]\left\{F_{unit}\right\}---\left(5\right)$

Utilize the displacement phase of each contact point that the displacement of each contact point that gear wheel Model I obtains obtains with gear wheel model- Subtract, i.e. formula (6), obtain this contact line gear wheel flank of tooth normal direction flexibility matrix R_G:

R_G[i, j]=d_G1-d_G2, (i, j=1 ..., m). (6).